It is proved that charged black-hole spacetimes of the composed Einstein-Born-Infeld field theory may develop superradiant instabilities when coupled to linearized charged massive scalar fields. In particular, using analytical techniques we prove that, for a given value $\bar{\beta }\ll 1$ of the dimensionless coupling parameter that characterizes the non-linear electromagnetic field theory, near-extremal Einstein-Born-Infeld black holes that satisfy the dimensionless inequalities $1-{\bar{\beta }}^2/225<Q/Q^{\mathrm{\text{extremal}}}<1$ lie in a sufficient region for superradiant instability to perturbations of charged massive scalar fields in the large-mass $\mu r_{\mathrm{\text{H}}}\gg 1$ regime. In addition, we prove that in the strong-coupling $\bar{\beta }\gg 1$ regime of the non-linear Born-Infeld field theory, charged black holes in the dimensionless regime ${{48\pi}/{\Gamma^4\big({1\over4}\big)}}<Q/Q^{\text{extremal}}<1$ lie in a sufficient region for superradiant instability when coupled to large-mass charged scalar fields. [Here $\{r_{\mathrm{\text{H}}},Q\}$ are respectively the horizon radius and the electric charge of the black hole, $Q^{\mathrm{\text{extremal}}}(\bar{\beta })$ is the $\bar{\beta }$-dependent charge parameter that characterizes the extremal (maximally-charged) Born-Infeld black hole, and $\mu$ is the proper mass of the scalar field].